Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T23:01:15.422Z Has data issue: false hasContentIssue false
  • English
  • Français

A Projected Algebraic Multigrid Method for Linear Complementarity Problems

Published online by Cambridge University Press:  28 May 2015

Jari Toivanen  and
Cornelis W. Oosterlee
Jari Toivanen*
Affiliation:
Institute for Computational and Mathematical Engineering, Durand Building, room 023A, Stanford University, Stanford, CA 94305, USA and Department of Mathematical Information Technology, Agora, FI-40014 University ofJyväskylä, Finland
Cornelis W. Oosterlee
Affiliation:
Centrum voor Wiskunde en Informatica, Modelling, Analysis and Simulation (MAS2), Kruislaan 413, Amsterdam, The Netherlands and Delft University of Technology, Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD Delft, The Netherlands
*
Corresponding author.Email address:toivanen@stanford.edu
Get access

Abstract

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial engineering.

Keywords

65M5565F9991G20Linear complementarity problemalgebraic multigridAmerican optionselasto-plastic torsion problem
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 1 , February 2012 , pp. 85 - 98
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Achdou, Y. and Pironneau, O., Computational Methods for Option Pricing, vol. 30 of Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, 2005.Google Scholar
[2] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), pp. 333–390.CrossRefGoogle Scholar
[3] Brandt, A. and Cryer, C. W., Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems, SIAM J. Sci. Statist. Comput., 4 (1983), pp. 655–684.CrossRefGoogle Scholar
[4] Briggs, W. L., Henson, V. E., and Mccormick, S. F., A multigrid tutorial, SIAM, Philadelphia, PA, second ed., 2000.CrossRefGoogle Scholar
[5] Clarke, N. and Parrott, K., Multigrid for American option pricing with stochastic volatility, Appl. Math. Finance, 6 (1999), pp. 177–195.CrossRefGoogle Scholar
[6] Cleary, A. J., Falgout, R. D., Henson, V. E., Jones, J. E., Manteuffel, T. A., Mccormick, S. F., Miranda, G. N., and Ruge, J. W., Robustness and scalability of algebraic multigrid, SIAM J. Sci. Comput., 21 (2000), pp. 1886–1908.CrossRefGoogle Scholar
[7] Cryer, C. W., The solution of a quadratic programming problem using systematic overrelaxation, SIAM J. Control, 9 (1971), pp. 385–392.CrossRefGoogle Scholar
[8] De Sterck, H., Yang, U. Meier, and Heys, J. J., Reducing complexity in parallel algebraic multi-grid preconditioners, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 1019–1039.CrossRefGoogle Scholar
[9] Elliott, C. M. and Ockendon, J. R., Weak and variational methods for moving boundary problems, vol. 59 of Research Notes in Mathematics, Pitman, Boston, MA, 1982.Google Scholar
[10] V Henson, E. and Yang, U. Meier, BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), pp. 155–177.CrossRefGoogle Scholar
[11] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financial Stud., 6 (1993), pp. 327–343.CrossRefGoogle Scholar
[12] Hoppe, R. and Kornhuber, R., Multi-grid methods for two phase Stefan problem, Tech. Report 171, TU Berlin, 1987.Google Scholar
[13] Hoppe, R. H. W., Multigrid algorithms for variational inequalities, SIAM J. Numer. Anal., 24 (1987), pp. 1046–1065.CrossRefGoogle Scholar
[14] Huang, J. and Pang, J.-S., Option pricing and linear complementarity, J. Comput. Finance, 2 (1998), pp. 31–60.CrossRefGoogle Scholar
[15] Ikonen, S. and Toivanen, J., Efficient numerical methods for pricing American options under stochastic volatility, Numer. Methods Partial Differential Equations, 24 (2008), pp. 104–126.CrossRefGoogle Scholar
[16], Operator splitting methods for pricing American options under stochastic volatility, Nu mer. Math., 113 (2009), pp. 299–324.CrossRefGoogle Scholar
[17] Ito, K. and Toivanen, J., Lagrange multiplier approach with optimized finite difference stencils for pricing American options under stochastic volatility, SIAM J. Sci. Comput., 31 (2009), pp. 2646–2664.CrossRefGoogle Scholar
[18] Jaillet, P., Lamberton, D., and Lapeyre, B., Variational inequalities and the pricing of American options, Acta Appl. Math., 21 (1990), pp. 263–289.CrossRefGoogle Scholar
[19] Kornhuber, R., Monotone multigrid methods for elliptic variational inequalities. I, Numer. Math., 69 (1994), pp. 167–184.CrossRefGoogle Scholar
[20] Krechel, A. and StÜben, K., Operator dependent interpolation in algebraic multigrid, in Multi-grid methods V, vol. 3 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 1998, pp. 189–211.Google Scholar
[21] Krechel, A., Parallel algebraic multigrid based on subdomain blocking, Parallel Comput., 27 (2001), pp. 1009–1031.CrossRefGoogle Scholar
[22] Oosterlee, C. W., On multigrid for linear complementarity problems with application to American-style options, Electron. Trans. Numer. Anal., 15 (2003), pp. 165–185.Google Scholar
[23] Rannacher, R., Finite element solution of diffusion problems with irregular data, Numer. Math., 43 (1984), pp. 309–327.CrossRefGoogle Scholar
[24] Reisinger, C. and Wittum, G., On multigrid for anisotropic equations and variational inequalities: Pricing multi-dimensional European and American options, Comput. Vis. Sci., 7 (2004), pp. 189–197.CrossRefGoogle Scholar
[25] Ruge, J. W. and StÜben, K., Algebraic multigrid, in Multigrid methods, vol. 3 of Frontiers Appl. Math., SIAM, Philadelphia, PA, 1987, pp. 73–130.Google Scholar
[26] StÜben, K., Algebraic multigrid: An introduction with applications, in Multigrid, Academic Press Inc., San Diego, CA, 2001.Google Scholar
[27] Trottenberg, U., Oosterlee, C. W., and SchÜller, A., Multigrid, Academic Press Inc., San Diego, CA, 2001.Google Scholar
[28] Wilmott, P., Dewynne, J., and Howison, S., Option pricing: mathematical models and computation, Oxford Financial Press, Oxford, 1993.Google Scholar
[29] Zvan, R., Forsyth, P. A., and Vetzal, K. R., Penalty methods for American options with stochastic volatility, J. Comput. Appl. Math., 91 (1998), pp. 199–218.CrossRefGoogle Scholar